An Average is a single number expressing a set of data. It is calculated by dividing the sum of the values in the set by their number, also called arithmetic mean. The basic formula for the average of n numbers x1, x2,……xn isÂ
(Average)A = (x1 + x2 + …….. + xn)/n
Average is the another name of mean and average in mathematics is used for finding and defining various values. In this article, we will learn about the average definition, average formula, average solved examples, and others in detail.
What is Average in Maths?
Average in mathematics is defined as the central value of the given data set. It is the ratio of the sum of all the values by the number of the values. For n terms, its average is given by first taking the sum of n numbers and then dividing them by n.
Average Definition
Average is defined as the value obtained by the dividing the sum of data by the given number of data.
The image below shows three rows of apples with 6, 11, and 7 apples and if we take the average of all three rows then we get 8 apples in each row.
Average Symbol
Average is just the mean of numbers and it is denoted xÌ„ (read as “x bar”). We also use the Greek letter “μ” to denote the average.
The average in mathematics is calculated using the formula sum of values divided by the number of values. Hence, the average formula is given as
Average = Sum of Values/Number of Values
For given n numbers x1, x2, x3 ,….., xn the average is given by the formula,
Average = (x1 + x2 + … + xn)/n
How to Calculate Average?
Study the following steps to find the average of various numbers
Step 1: Note all the observations and find the total number of observations (say n)
Step 2: Find the sum of all observations.
Step 3: Divide the sum obtained in Step 2 by the number of observations (n)
Step 4: Simplify to obtain the required value of Average.
Example: Find the average of 3, 4, 7, 8, 10, and 12.
Solution:
Given values,
Number of Observations = 6
Sum of Observations = 3 + 4 + 7 + 8 + 10 + 12 = 34
Average = 34/6 = 5.67
What is Average Used For?
Average is used to represent a large amount of data with a single number. It helps us find the central value of a data set.
Some practical applications of average are:
- Calculating the average time of commute to work or school can help you plan your schedule.
- Calculating the batting average in cricket helps in assessing batsman performance.
- Calculating average customer reviews before buying new things.
- Calculating average household income, average unemployment rate, or average inflation rate to understand economic trends.
- Calculating average daily sales of a product, to stock the right amount.
What is Mean?
Mean in mathematics is the measure of central tendency. It is also called average or arithmetic mean. It is calculated by dividing the sum of values by the total number of values.
There are three types of mean in mathematics that are,
- Arithmetic Mean
- Geometric Mean
- Harmonic Mean
Now let’s learn about them in detail.
Also Read: Mean, Median and Mode
Arithmetic Mean
The arithmetic mean is another name for the average. It is the sum of values divided by the number of values. The formula to calculate the arithmetic mean for n values x1, x2, …, xn is,
A.M. = (n1 + n2 + n3 + n4 + … + nn)/n
Geometric Mean
Geometric Mean is one of the measures of the central tendency. It is calculated by taking the nth root of the product of all the given numbers. The formula to calculate the geometric mean for n values x1, x2, …, xn is,
G.M. = n√(x1.x2…xn)
Harmonic Mean
The harmonic mean is one of the Pythagorean means other than the Arithmetic Mean and Geometric Mean. It is calculated by dividing the number of the reciprocal by the sum of the reciprocal values. The harmonic mean is always lower as compared to the geometric and arithmetic mean.
The formula to calculate the harmonic mean for n values x1, x2, …, xn is,
H.M. = n/{(1/x1) + (1/x2) + … + (1/xn)}
Average of Negative Numbers
The average of the negative number is simply calculated by taking the sum of the observations divided by the number of the observations. Negative numbers have no effect in finding the average of the negative numbers. This is explained by the example,
Example: Find the average of -8, -4, 0, 4, 8
Solution:
Given,
Number of Observations = 5
Sum of Observations = (-8) + (-4) + 0 + 4 + 8 = 0
Average = 0/5 = 0
Average of Two Numbers
The average of two numbers is simply the sum of two numbers divided by 2. Suppose we are given two numbers ‘a’ and ‘b’ then their average is calculated as,
Average = (a+b)/2
Example: Find the average value of 80 and 100
Solution:
Given,
Average = (a+b)/2
= (80+100)/2 = 180/2
= 90
Some of the important tips and tricks to solve average questions are mentioned below. These formulas will help students and will be useful in boards and competitive exams.
Average of first n natural numbers:
- Sum of first n natural numbers = n(n + 1)/2
- Average of first n natural numbers = (n + 1)/2
Average of first n natural number squares,
- Sum of square of first n natural numbers = n(n+1)(2n+1)/6
- Average of square of first n natural numbers = (n+1)(2n+1)/6
Average of first n natural number cubes:
- Sum of cube of first n natural numbers = [n(n+1)/2]2
- Average of cube of first n natural numbers = n[(n+1)/2]2
Average of first n natural odd numbers:
- Sum of first n natural odd numbers = n2
- Average of first n natural odd number = n
Average of first n natural even numbers:
- Sum of first n natural even numbers = n(n+1)
- Average of first n natural even numbers = n + 1
Read More,
Examples on Average
Here are some numerical examples on average with solutions. These solved examples will help students understand and practice the concept of average.
Example 1: Find the average of the square of the first 16 natural numbers.
Solution:
Sum of square of first n natural number = n(n+1)(2n+1)/6Â
Avg. of square of first n natural number = (n+1)(2n+1)/6Â
Average = (16+1)(2×16+1)/6Â
= 17 × 33 /6Â
= 187/2Â
Example 2: The average of 9 observations is 87. If the average of the first five observations is 79 and the average of the next three is 92. Find the 9th observation.Â
Solution:
Average of 9 observations = 87Â
Sum of 9 observations = 87 × 9 = 783Â
Average of first 5 observations = 79Â
Sum of first 5 observations = 79 × 5 = 395Â
Sum of 6th,7th and 8th = 92 × 3 = 276Â
9th number = 783 – 395 – 276 = 112Â
Example 3: Five years ago the average of the Husband and wife was 25 years, today the average age of the Husband, wife, and child is 21 years. How old is the child?Â
Solution:
H + W = 25Â
Sum of ages of both 5 years before = 25×2 = 50Â
Today, sum of their ages is = 50 + 5 + 5 = 60Â
Today avg. of H + W + C = 21Â
Sum of ages of H , W and C = 21×3 = 63Â
Age of child = 63 – 60 = 3 years
Example 4: There are 42 students in a hostel. If the number of students increased by 14. The expense of mess increased by Rs 28 per day. While the average expenditure per head decreased by Rs 2. Find the original expenditure.Â
Solution:
Total students after increment = 42 + 14 = 56Â
Let the expenditure of students is A Rs/day.Â
Increase in expenditure Rs 28/day.Â
Acc. to questionÂ
42A + 28 = 56(A – 2)Â
42A + 28 = 56A – 112Â
14A = 140Â
A = 10Â
Hence, the original expenditure of the student was Rs 10/day.Â
Example 5: The average of 200 numbers is 96 but it was found that 2 numbers 16 and 43 are mistakenly calculated as 61 and 34. Find his correct average it was also found that the total number is only 190.Â
Solution:
Average of 200 numbers = 96Â
Sum of 200 numbers = 96 x 200 = 19200Â
Two numbers mistakenly calculated as 61 and 34 instead of 16 and 43.Â
So, 61 + 34 = 95Â
16 + 43 = 59Â
Diff = 95 – 59 = 36Â
So, Actual sum of 200 numbers = 19200 – 36 = 19164Â
Total numbers are also 190 instead of 200
So, correct average = 19164/190 = 100.86
Example 6: A batsman scored 120 runs in his 16th innings due to this his average increased by 5 runs. Find his current average.Â
Solution:
Let the average of 15 innings is A
Acc. to questionÂ
15A + 120 = 16(A + 5)Â
15A + 120 = 16A + 80Â
A = 40Â
Hence, current average of the batsman is (40 + 5) = 45Â
Example 7: There are three natural numbers if the average of any two numbers is added with the third number 48,40 and 36 will be obtained. Find all the natural numbers.Â
Solution:
Let a, b and c are the numbers
GivenÂ
=> a + b + 2c = 96 ………(1)Â
(b+c)/2 + a = 40Â
=> 2a + b + c = 80 ……….(2)Â
(c+a)/2 + b = 36Â
=> a + 2b + c = 72 ……….(3)Â
Add (1)(2)(3), we getÂ
4(a + b + c) = 248Â
a + b + c = 62Â
From 1, 2, and 3
(a+b+c) + c = 96Â
62 + c = 96Â
a + (a+b+c) = 80Â
a + 62 = 80Â
b + (a+b+c) = 72Â
b + 62 = 72Â
Example 8: A biker travels at a speed of 60 km/hr from A to B and returns at a speed of 40 km/hr. What is the average speed of the total journey?Â
Solution:
Let a is the distance between A and B
Total distance travel in journey = 2aÂ
Time to travel from A to B = Distance/speed = a/60Â
Time to travel from B to A = Distance/speed = a/40Â
Total time of journey = a/60 + a/40Â
Average speed = Total distance/Total timeÂ
=2a / (a/60 + a/40)Â
=240 × 2a /10aÂ
= 240/5Â
= 48Â
Hence, the average speed is 48 km/hr.
Practice Questions on Average
Below are some practice questions of average. Students should try solving these questions and test their skills. Look at the solved examples above, if stuck on a problem.
Q1. Average temp. of Monday, Tuesday, Wednesday, and Thursday are 31°, and the average temp. of Tuesday, Wednesday, Thursday, and Friday are 29.5°. If the temp of Friday is 4/5 times of Monday. Find the temp for Monday.
Q2. The average age of boys in school is 13 years and of girls is 12 years. If the total number of boys is 240, then find the number of girls if the average of school is 12 years 8 months.
Q3. If the runs scored by a batsman in 5 matches are 56, 102, 23, 45, and 78. Find the average run scored by him.
FAQs on Average
What is Average?
Average in mathematics is defined as the mean of two numbers. It is used to find the central value of the data set. It is calculated by taking the ratio of all the observation by number of observations.
What is Average Formula?
The formula to calculate the average of numbers is,
Average = (Sum of Terms)/ (Number of Terms)
What is Average of First n numbers?
The average of first n natural is calculated below,
Sum of n natural number = n(n + 1)/2
- Average of n natural number = n(n + 1)/2n = (n + 1)/2
What are the three types of Averages?
The three types of average are: mean, median and mode.
Is Average and Mean the Same?
Yes, average in mathematics is similar to the mean in mathematics. It is calculated by dividing the sum of vales by number values.
What is Weighted Average?
Weighted Average is average of the dataset that takes the weight of each data in a dataset
How to Calculate Weighted Average?
We can calculate weighted average by taking sum of product of each data with their weight and then dividing the total weight by total number of data
Last Updated :
08 Mar, 2024
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